MATHEMATICS
UPSC Mathematics MCQs
1:-Find a simpler expression for where S and R are any two sets and U is the Universal Set.
A:-S
B:-R
C:-U
D:-`phi`
Ans: C
2:-Fibonacci number `f_(n)=f_(n-1)+f_(n-2)` where `f_1=1` and `f_2=1` is even if n is divisible by :``
A:-2
B:-3
C:-multiples of 2
D:-None of these
Ans: B
3:-If x, y and z are three elements in a Boolean algebra such that `x=yz^1+y^1z,` then write an expression for `x^1`.
A:-`(y+z) (y^1+z^1)`
B:-`yz+y^1 z^1`
C:-`y^1+z^1`
D:-None of these
Ans: B
4:-The domain of the function is :
A:-( - `oo`, - 1) `uu` (1, `oo`)
B:-( - `oo`, - 2) `uu` (2, `oo`)
C:-( - 2, - 1) `uu` (1, 2)
D:-None of these
Ans: C
5:-The number of vertices of a 2-regular graph with 16 edges is :
A:-16
B:-32
C:-8
D:-`16^2`
Ans: A
6:-Which of the following is not true `K_3,``_3` is a :
A:-Bipartite graph
B:-Complete graph
C:-Cubic graph
D:-3 regular graph
Ans: B
7:-For any vector space V which of the following set is a subspace having a single element :
A:-{0}
B:-{`phi`}
C:-`phi`
D:-{{0}}
Ans: A
A:-S
B:-R
C:-U
D:-`phi`
Ans: C
2:-Fibonacci number `f_(n)=f_(n-1)+f_(n-2)` where `f_1=1` and `f_2=1` is even if n is divisible by :``
A:-2
B:-3
C:-multiples of 2
D:-None of these
Ans: B
3:-If x, y and z are three elements in a Boolean algebra such that `x=yz^1+y^1z,` then write an expression for `x^1`.
A:-`(y+z) (y^1+z^1)`
B:-`yz+y^1 z^1`
C:-`y^1+z^1`
D:-None of these
Ans: B
4:-The domain of the function is :
A:-( - `oo`, - 1) `uu` (1, `oo`)
B:-( - `oo`, - 2) `uu` (2, `oo`)
C:-( - 2, - 1) `uu` (1, 2)
D:-None of these
Ans: C
5:-The number of vertices of a 2-regular graph with 16 edges is :
A:-16
B:-32
C:-8
D:-`16^2`
Ans: A
6:-Which of the following is not true `K_3,``_3` is a :
A:-Bipartite graph
B:-Complete graph
C:-Cubic graph
D:-3 regular graph
Ans: B
7:-For any vector space V which of the following set is a subspace having a single element :
A:-{0}
B:-{`phi`}
C:-`phi`
D:-{{0}}
Ans: A
8. (x % of y) + (y % of x) is equivalent to .
(A) 2 % of xy (B) 2 % of (xy/100) (C) xy % of 100 (D) 100 % of xy
Ans: A
9. The sum of the digits of a two digit number is 12. If the new number formed by reversing the digits is greater than the original number by 54, find the original number.
(A) 39 (B) 57 (C) 66 (D) 93
Ans: A
(A) 2 % of xy (B) 2 % of (xy/100) (C) xy % of 100 (D) 100 % of xy
Ans: A
9. The sum of the digits of a two digit number is 12. If the new number formed by reversing the digits is greater than the original number by 54, find the original number.
(A) 39 (B) 57 (C) 66 (D) 93
Ans: A
10:-The number of tangents than can be drawn from (0, 0) to the circle `x^(2)+y^(2)-2x-4y-4=0` is
A:-0 B:-1
C:-2 D:-None of these
Ans: A
11:-The total number of terminating zeros in 100! is
A:-20 B:-21
C:-24 D:-None of these
Ans: C
12:-If `1^(st)` January 2018 is Monday, what will be 2036 January `1^(st)` ?
A:-Monday B:-Tuesday
C:-Wednesday D:-None of these
Ans: B
13:-For a Boolean Algebra <X, +, .,'> where X = {0, 1, x, y}; the value of x + y is
A:-0 B:-x
C:-y D:-1
Ans: D
14:-Let X be a Boolean Algebra. Then the number of elements in X cannot be
A:-2 B:-4
C:-6 D:-8
Ans: C
15:-Which term is zero in the arithmetic progression –123, –120, –117, .........
A:-100 B:-52 C:-41 D:-42 Ans: D
16:-In an examination 92% of the candidates passed and 110 failed. How many candidates appeared for the examination ?
A:-1200 B:-1275 C:-1375 D:-1100
Ans: C
17:-A wire 88 cm long is bent into a circle. What is the area enclosed within it ? Use (`Pi`= `22/7` )
A:-616 `cm^(2)` B:-661 `cm^(2)` C:-366 `cm^(2)` D:-132 `cm^(2)`
Ans: A
18:-If F is a field, then the number of elements in F cannot be
A:-3 B:-5
C:-15 D:-125
Ans: C
19:-Which of the following is false about a field F of 81 elements ?
A:-F has a subfield of 27 elements B:-F has a subfield of 9 elements
C:-F has a subfield of 3 elements D:-F has exactly 3 subfields including F
Ans: A
20:-Let <F, +,.> be a field of 16 elements. Then <F, +> is isomorphic to
A:-`ZZ_(16)` B:-`ZZ_(2)` ×`ZZ_(2)` ×`ZZ_(4)`
C:-`ZZ_(2)` ×`ZZ_(8)` D:-`ZZ_(2)` ×`ZZ_(2)` ×`ZZ_(2)` ×`ZZ_(2)`
Ans: D
21:-Let G be a non-abelian group. Then order of G can be
A:-25 B:-35
C:-55 D:-255
Ans: C
22:-Let G = `SL_(4)` (`ZZ_(3)`) ` ` , the group of all 4×4 matrices over `ZZ_(3)` with determinant 1. Then the order of any of its Sylow 3-subgroup is
A:-`3^(5)` B:-`3^(6)`
C:-`3^(7)` D:-None of these
Ans: B
23:-Which of the following is not constructible ?
A:-20-gon B:-30-gon
C:-50-gon D:-60-gon
Ans: C
24:-Which of the following is false ?
A:-There exists a vector space of 81 elements
B:-There exists a vector space of 81 elements over a field of 3 elements
C:-There exists a vector space of 81 elements over a field of 9 elements
D:-There exists a vector space of 81 elements over a field of 27 elements
Ans: D
25:-Which of the following linear transformation is invertible ?
A:-T(x, y) = (2x + y, x + `(1)/(2)`y) B:-T(x, y) = (2x +y, `(1)/(2)`x + y)
C:-T(x, y) = (x + `(1)/(2)`y, 2x + y) D:-T(x, y) = (x + y, x + y)
Ans: B
26:-If the characteristic polynomial of the linear transformation T : `RR^(9)` `->RR^(9)` is `x^(9)` + 4x + 1, then det (T− I) is
A:-−6 B:-−9
C:-−1 D:-1
Ans: A
27:-Which of the following is false ?
A:-A − B is self adjoint if A and B are so B:-Every unitary operator is normal
C:-Every normal operator is self adjoint D:-A + B is self adjoint if A and B are so
Ans: C
28:-Which of the following is a Hilbert space ?
A:-`l^(1)` B:-`l^(2)`
C:-`l^(oo)` D:-None of these
Ans: B
29:-Let f(x, y)= `{((x^(3))/(x^(3)+y^(2)) if (x, y)!=(0, 0)), (0 if (x, y)=(0, 0)):}` Then which is not true ?
A:-f is continuous at (0, 0)
B:-Partial derivatives exists at (0, 0)
C:-Directional derivatives exists at (0, 0)
D:-Partial derivatives are not bounded functions on `RR^(2)`
Ans: D
30:-If the vectors i + 2j + 3k, 4i + 5j + 6k and 5i + mj + 9k are coplanar, then the value of m is
A:-7 B:-6
C:-5 D:-None of these
Ans: A
31:-Let `lim_(x->0)f(x)/x=l`, where 0< `l` `<oo`. Then`lim_(x->0)` `f(x)`
A:-Is always 0 B:-Need not exists
C:-Exists, but not always zero D:-Exists and depends on `l`
Ans: A
32:-Which of the following function is not differentiable on `(0, (pi)/(2))` ?
A:-sin x B:-|sin x|
C:-max{sin x, cos x} D:-None of these
Ans: C
33:-Let A be a 5×4 matrix and B be a 4×5 matrix. Then 1 is necessarily an eigen value of
A:-AB B:-AB + I
C:-BA D:-BA + I
Ans: B
34:-The equation `x^(2)` +`y^(2)` + 2xy −1 = 0 represents
A:-Parabola B:-Ellipse
C:-Hyperbola D:-Pair of straight lines
Ans: D
35:-Which of the following is not a Banach space ?
A:-`C_(00)` B:-`l^(oo)`
C:-`l^(2)` D:-None of these
Ans: A
36:-Which is false about `NN` ?
A:-`NN` cannot be written as a denumerable union of denumerable disjoint sets
B:-`NN` is well ordered
C:-`NN` is a closed subset of `RR`
D:-None of these
Ans: A
37:-E = {p`in` `QQ` :2<`p^(2)` <3}. Then which is false ?
A:-E is open in `QQ` B:-E is closed in `QQ`
C:-E is a bounded subset of `QQ` D:-E is a compact subset of `QQ`
Ans: D
38:-Which of the following is not uniformly continuous on `RR` ?
A:-x B:-sin x
C:-x sin x D:-`(1)/(1+x^(2))`
Ans: C
39:-Which is false ?
A:-There exists a function f : `RR` `->` `RR` which is continuous exactly at one point
B:-There exists a function f :`RR` `->RR` which is differentiable exactly at one point
C:-There exists a function f :`RR` `->` `RR` which is discontinuous everywhere
D:-None of these
Ans: D
40:-If `int_(1)^(oo)` f(x) dx = L where 0 < L < `oo`; then `lim_(x->oo)` f(x) is
A:-0 B:-`oo` ``
C:-always exists D:-need not exists
Ans: D
41:-Which is false ?
A:-cos z = 5 has a solution in `CC` B:-sin z is a polynomial in `CC`
C:-`sin^(z)` +`cos z^(2)` = 1 for all z `in` `CC` D:-Every non constant polynomial in `CC`[x] has a zero in `CC`
Ans: B
42:-The image of the line y = 1 under the mapping w = sin z is
A:-Parabola B:-Ellipse C:-Rectangular Hyperbola D:-None of these
Ans: B
43:-Consider the metric space (`NN`, d) where d is given by d(m, n) = |`1/m` − `1/n`| . Then which is false ?
A:-d is a bounded metric on `NN` B:-d induces the discrete topology on `NN`
C:-`{1/n}` converges to 0 D:-This space is Hausdorff
Ans: C
44:-Which is not a productive property ?
A:-Connectedness B:-Compactness
C:-Locally connectedness D:-Path connectedness
Ans: C
45:-Let X = (0, 1) and Y = `RR` . Then which is true ?
A:-X and Y are the same as metric spaces B:-X and Y are the same as topological spaces
C:-both (a) and (b) D:-Neither (a) nor (b)
Ans: B
46:-Which of the following topological property is not preserved under a continuous function ?
A:-Connectedness B:-Compactness
C:-First countability D:-None of these
Ans: C
47:-Which is true ?
A:-On `RR` co-finite topology is weaker than usual topology
B:-On `RR` usual topology is weaker than co-finite topology
C:-On `RR` co-finite topology and usual topology are not comparable
D:-On `RR` co-finite topology and usual topology are the same
Ans: A
48:-If every closed interval [a, b] with a < b is open with respect to some topology on `RR` ; then with respect to this topology, closure of [27, 37] is
A:-[27, 37] B:-(−`oo` , 37]
C:-[27, `oo` ) D:-`RR`
Ans: A
49:-For the space `RR` with co-countable topology, which is false ?
A:-Uniqueness of limits exists in this space for the convergence of sequences
B:-This space is not Hausdorff
C:-{`1/n`} is divergent in this space` `
D:-None of these
Ans: D
50:-If f : `RR` `->` `RR` is a twice differentiable function with `lim_(x->oo)` (2f (x) + 3f’(x) + f”(x)) = 0 with `lim_(x->oo)` f(x) = `lim_(x->oo)` f’(x) = 1; then `lim_(x->oo)` f”(x) is
A:-0 B:-1 C:-10 D:-11
Ans: D
A:-0 B:-1
C:-2 D:-None of these
Ans: A
11:-The total number of terminating zeros in 100! is
A:-20 B:-21
C:-24 D:-None of these
Ans: C
12:-If `1^(st)` January 2018 is Monday, what will be 2036 January `1^(st)` ?
A:-Monday B:-Tuesday
C:-Wednesday D:-None of these
Ans: B
13:-For a Boolean Algebra <X, +, .,'> where X = {0, 1, x, y}; the value of x + y is
A:-0 B:-x
C:-y D:-1
Ans: D
14:-Let X be a Boolean Algebra. Then the number of elements in X cannot be
A:-2 B:-4
C:-6 D:-8
Ans: C
15:-Which term is zero in the arithmetic progression –123, –120, –117, .........
A:-100 B:-52 C:-41 D:-42 Ans: D
16:-In an examination 92% of the candidates passed and 110 failed. How many candidates appeared for the examination ?
A:-1200 B:-1275 C:-1375 D:-1100
Ans: C
17:-A wire 88 cm long is bent into a circle. What is the area enclosed within it ? Use (`Pi`= `22/7` )
A:-616 `cm^(2)` B:-661 `cm^(2)` C:-366 `cm^(2)` D:-132 `cm^(2)`
Ans: A
18:-If F is a field, then the number of elements in F cannot be
A:-3 B:-5
C:-15 D:-125
Ans: C
19:-Which of the following is false about a field F of 81 elements ?
A:-F has a subfield of 27 elements B:-F has a subfield of 9 elements
C:-F has a subfield of 3 elements D:-F has exactly 3 subfields including F
Ans: A
20:-Let <F, +,.> be a field of 16 elements. Then <F, +> is isomorphic to
A:-`ZZ_(16)` B:-`ZZ_(2)` ×`ZZ_(2)` ×`ZZ_(4)`
C:-`ZZ_(2)` ×`ZZ_(8)` D:-`ZZ_(2)` ×`ZZ_(2)` ×`ZZ_(2)` ×`ZZ_(2)`
Ans: D
21:-Let G be a non-abelian group. Then order of G can be
A:-25 B:-35
C:-55 D:-255
Ans: C
22:-Let G = `SL_(4)` (`ZZ_(3)`) ` ` , the group of all 4×4 matrices over `ZZ_(3)` with determinant 1. Then the order of any of its Sylow 3-subgroup is
A:-`3^(5)` B:-`3^(6)`
C:-`3^(7)` D:-None of these
Ans: B
23:-Which of the following is not constructible ?
A:-20-gon B:-30-gon
C:-50-gon D:-60-gon
Ans: C
24:-Which of the following is false ?
A:-There exists a vector space of 81 elements
B:-There exists a vector space of 81 elements over a field of 3 elements
C:-There exists a vector space of 81 elements over a field of 9 elements
D:-There exists a vector space of 81 elements over a field of 27 elements
Ans: D
25:-Which of the following linear transformation is invertible ?
A:-T(x, y) = (2x + y, x + `(1)/(2)`y) B:-T(x, y) = (2x +y, `(1)/(2)`x + y)
C:-T(x, y) = (x + `(1)/(2)`y, 2x + y) D:-T(x, y) = (x + y, x + y)
Ans: B
26:-If the characteristic polynomial of the linear transformation T : `RR^(9)` `->RR^(9)` is `x^(9)` + 4x + 1, then det (T− I) is
A:-−6 B:-−9
C:-−1 D:-1
Ans: A
27:-Which of the following is false ?
A:-A − B is self adjoint if A and B are so B:-Every unitary operator is normal
C:-Every normal operator is self adjoint D:-A + B is self adjoint if A and B are so
Ans: C
28:-Which of the following is a Hilbert space ?
A:-`l^(1)` B:-`l^(2)`
C:-`l^(oo)` D:-None of these
Ans: B
29:-Let f(x, y)= `{((x^(3))/(x^(3)+y^(2)) if (x, y)!=(0, 0)), (0 if (x, y)=(0, 0)):}` Then which is not true ?
A:-f is continuous at (0, 0)
B:-Partial derivatives exists at (0, 0)
C:-Directional derivatives exists at (0, 0)
D:-Partial derivatives are not bounded functions on `RR^(2)`
Ans: D
30:-If the vectors i + 2j + 3k, 4i + 5j + 6k and 5i + mj + 9k are coplanar, then the value of m is
A:-7 B:-6
C:-5 D:-None of these
Ans: A
31:-Let `lim_(x->0)f(x)/x=l`, where 0< `l` `<oo`. Then`lim_(x->0)` `f(x)`
A:-Is always 0 B:-Need not exists
C:-Exists, but not always zero D:-Exists and depends on `l`
Ans: A
32:-Which of the following function is not differentiable on `(0, (pi)/(2))` ?
A:-sin x B:-|sin x|
C:-max{sin x, cos x} D:-None of these
Ans: C
33:-Let A be a 5×4 matrix and B be a 4×5 matrix. Then 1 is necessarily an eigen value of
A:-AB B:-AB + I
C:-BA D:-BA + I
Ans: B
34:-The equation `x^(2)` +`y^(2)` + 2xy −1 = 0 represents
A:-Parabola B:-Ellipse
C:-Hyperbola D:-Pair of straight lines
Ans: D
35:-Which of the following is not a Banach space ?
A:-`C_(00)` B:-`l^(oo)`
C:-`l^(2)` D:-None of these
Ans: A
36:-Which is false about `NN` ?
A:-`NN` cannot be written as a denumerable union of denumerable disjoint sets
B:-`NN` is well ordered
C:-`NN` is a closed subset of `RR`
D:-None of these
Ans: A
37:-E = {p`in` `QQ` :2<`p^(2)` <3}. Then which is false ?
A:-E is open in `QQ` B:-E is closed in `QQ`
C:-E is a bounded subset of `QQ` D:-E is a compact subset of `QQ`
Ans: D
38:-Which of the following is not uniformly continuous on `RR` ?
A:-x B:-sin x
C:-x sin x D:-`(1)/(1+x^(2))`
Ans: C
39:-Which is false ?
A:-There exists a function f : `RR` `->` `RR` which is continuous exactly at one point
B:-There exists a function f :`RR` `->RR` which is differentiable exactly at one point
C:-There exists a function f :`RR` `->` `RR` which is discontinuous everywhere
D:-None of these
Ans: D
40:-If `int_(1)^(oo)` f(x) dx = L where 0 < L < `oo`; then `lim_(x->oo)` f(x) is
A:-0 B:-`oo` ``
C:-always exists D:-need not exists
Ans: D
41:-Which is false ?
A:-cos z = 5 has a solution in `CC` B:-sin z is a polynomial in `CC`
C:-`sin^(z)` +`cos z^(2)` = 1 for all z `in` `CC` D:-Every non constant polynomial in `CC`[x] has a zero in `CC`
Ans: B
42:-The image of the line y = 1 under the mapping w = sin z is
A:-Parabola B:-Ellipse C:-Rectangular Hyperbola D:-None of these
Ans: B
43:-Consider the metric space (`NN`, d) where d is given by d(m, n) = |`1/m` − `1/n`| . Then which is false ?
A:-d is a bounded metric on `NN` B:-d induces the discrete topology on `NN`
C:-`{1/n}` converges to 0 D:-This space is Hausdorff
Ans: C
44:-Which is not a productive property ?
A:-Connectedness B:-Compactness
C:-Locally connectedness D:-Path connectedness
Ans: C
45:-Let X = (0, 1) and Y = `RR` . Then which is true ?
A:-X and Y are the same as metric spaces B:-X and Y are the same as topological spaces
C:-both (a) and (b) D:-Neither (a) nor (b)
Ans: B
46:-Which of the following topological property is not preserved under a continuous function ?
A:-Connectedness B:-Compactness
C:-First countability D:-None of these
Ans: C
47:-Which is true ?
A:-On `RR` co-finite topology is weaker than usual topology
B:-On `RR` usual topology is weaker than co-finite topology
C:-On `RR` co-finite topology and usual topology are not comparable
D:-On `RR` co-finite topology and usual topology are the same
Ans: A
48:-If every closed interval [a, b] with a < b is open with respect to some topology on `RR` ; then with respect to this topology, closure of [27, 37] is
A:-[27, 37] B:-(−`oo` , 37]
C:-[27, `oo` ) D:-`RR`
Ans: A
49:-For the space `RR` with co-countable topology, which is false ?
A:-Uniqueness of limits exists in this space for the convergence of sequences
B:-This space is not Hausdorff
C:-{`1/n`} is divergent in this space` `
D:-None of these
Ans: D
50:-If f : `RR` `->` `RR` is a twice differentiable function with `lim_(x->oo)` (2f (x) + 3f’(x) + f”(x)) = 0 with `lim_(x->oo)` f(x) = `lim_(x->oo)` f’(x) = 1; then `lim_(x->oo)` f”(x) is
A:-0 B:-1 C:-10 D:-11
Ans: D
51:-The third approximation `y_(3)` (x) for the I.V.P., y’ = 2x(1 + y); y(0) = 0 by Pickard's method is
A:-`x` + `x^(2)/2` + `x^(3)/6` B:-`x^(2)` + `x^(4)/2` + `x^(6)/6` C:-`x` + `x^(2)/2 + x^(8)/6` D:-None of these
Ans: B
52:-If F (a, b, c, x) is the hyper geometric series, then`lim_(b->oo)` F(a, b, a, `x/b` ) is
A:-`(1+x)^(a)` B:-log(1 + x) C:-`e^(x)` D:-sin x
Ans: C
53:-If `P_(n)(x)`is the Legendre Polynomial, then `P_(n)(-1)` is
A:-1 B:-− 1 C:-`(-1)^(n)` D:-0
Ans: C
54:-The equation `U_(xxxx_)+x^(2)U_(yy)` = 0 is
A:-elliptic B:-parabolic C:-hyperbolic D:-none of these
Ans: A
55:-If a, b, c are three constants such that `U_(x)` = a, `U_(y)` = b and `U_(z)` = c with u(x, y, z) = 0 at (0, 0, 0); then U(1, 0, 0) is
A:-a B:-b C:-c D:-none of these
Ans: A 8
56-If 0 < s < 1, then `int_0^oo(x^(s-1))/(1+x)dx` is
A:-`beta` (1, s) B:-`beta` (1− s, 1) C:-`beta` (1− s, s) D:-divergent
Ans: C
57:-The Laplace transform L {`(sin omegat)/(t)`} is
A:-`cot^(-1)((s)/(w))` B:-`(omega)/(s^(2)+omega^(2))` C:-`(s)/(s^(2)+omega^(2))` D:-none of these
Ans: A
58:-Which of the following is a periodic function on `RR` ?
A:-x −[x] B:-x C:-[x] D:-none of these
Ans: A
59:-Consider the fuzzy sets A and B on X = [0,`(pi)/(2)` ] where A(x) = sin x and B(x) = cos x. Then 0.8 − cut of A`nn`B is
A:-`Phi` B:-X
C:-An uncountable subset of X D:-A denumerable subset of X
Ans: A 9
60:-Which is false about the Cantor set ?
A:-Cantor set is perfect B:-Cantor set is compact
C:-Cantor set is a fractal D:-Cantor set is an uncountable set with positive Lebesgue measure
Ans: D
61:-The Voltra integral equation corresponding to the differential equation y” + xy = 1, y(0) = y’(0) = 0 is
A:-`y(x)` =`(x^(2))/(2)` −`int_0^xty(t)dt`
B:-`y(x)=x^(2)/2` −`int_0^x(x-t) y(t) dt`
C:-`y(x)=(x^(2))/(2)` −`int_0^(x)(x-t) ty (t) dt`
D:-`y(x)=(x)/(2)` −`int_0^x(x-t)ty(t) dt`
Ans: C
62:-For the Euler's gamma function Γ(z)Γ(1− z) is
A:-`(pi)/(cospiz)` B:-`(pi)/(sinpiz)` C:-`(pi)/(cotpiz)` D:-none of these
Ans: B
63:-Which of the following is not a maximal geodesic for the cylinder `x_(1)^(2)` + `x_(2)^(2)` = 1 in `RR^(3)` ?
A:-Vertical line B:-Horizontal circle C:-Helix D:-None of these
Ans: D
64:-Which of the following is reducible over `RR` ?
A:-`x^(2)` + 1 B:-`x^(2)` + 2 C:-`x^(5)` +`x^(3)+``x^(2)` + x + 1 D:-none of these
Ans: C
65. The average of 14, 11, 8, x, 19 is 13. The value of x is ______
(A) 52 (B) 12 (C) 16 (D) 13
Ans: D
66. If a : b = 2 : 3, b : c = 4 : 5, then a : b : c = ______
(A) 4 : 3 : 5 (B) 2 : 4 : 5 (C) 8 : 12 : 15 (D) 8 : 10 : 20
Ans: C
67. If 20% of a number is 140, then 16% of that number is _____
(A) 120 (B) 114 (C) 110 (D) 112
Ans: D
68. If by selling an article for Rs. 360, Ramu gains 20%. Find his cost price.
(A) 320 (B) 300 (C) 310 (D) 380
Ans: B
69:-16×8(6 − 4) ÷2+9= ___________
A:-219 B:-160 C:-137 D:-90
Ans: C
70:-20.4 × 0.44 ÷ 3.52 = ___________
A:-2.55 B:-3.55 C:-2.58 D:-5.01
Ans: A
71:-The average of 10 numbers is 30. If the average of 6 of these numbers is 29, what is the average of the remaining 4 numbers ? A:-31 B:-28.5 C:-30 D:-31.5
Ans: D
72:-The ratio of the length of a ground to its width is 5 : 2. Find its length if the width is 50 meters
A:-100 m B:-125 m C:-160 m D:-120 m
Ans: B
73:-A sum of money deposited at 2% per annum compounded annually becomes Rs. 10,404 at the end of 2 years. Find the sum deposited A:-Rs. 10,000 B:-Rs. 10,400 C:-Rs. 8,000 D:-Rs. 9,500
Ans: A
74:-40 men can do a piece of work in 20 days. If there are 50 men to do the same work how long will they take to finish it ?
A:-15 days B:-20 days C:-16 days D:-10 days
Ans: C
75:-A train running at 54 Km/hr crosses a telegraph post in 10 seconds. What is the length of the train ?
A:-540 m B:-150 m C:-100 m D:-75 m
Ans: B
76. A relation R = {(1, 1), (1, 2), (2, 1)} is defined on the set A = {1, 2, 3}, then relation is
(a) Reflexive (b) Reflexive and transitive (c) Equivalence relation (d) Symmetric
Ans: D
77. The number of generators of a cyclic group of order 12 is
(a) 4 (b) 3 (c) 2 (d) 1
Ans: A
78. 20 persons were invited for a party. What is the number of ways in which they and the host can be seated at a circular table such that two particular persons be seated on either side of the host ?
(a) 20 ! (b) 19 ! (c) 2(18 !) (d) (18 !)
Ans: C
79. The two positive numbers, whose difference is 12 and whose A.M. exceeds their G.M. by 2, are
(a) 32, 20 (b) 25, 13 (c) 20, 8 (d) 16, 4
Ans: D
80. How many unrelated conditions are required to determine a plane ?
(a) 4 (b) 3 (c) 2 (d) 1
Ans: B
81. Which of the following is the best measure of dispersion ?
(a) Range (b) Mean deviation (c) Standard deviation (d) Co-efficient of variation
Ans: C
82. Probability that a leap year contains 53 Sundays is
(a) 1/ 7
(b) 2 /7
(c) 3/ 7
(d) 4/ 7
Ans: B
83. The relationship between mean deviation (M.D.) and the standard deviation (S.D.) in the normal distribution, is approximately
(a) 3 M.D. = 2 S.D. (b) 5 M.D. = 4 S.D. (c) 6 M.D. = 5 S.D. (d) M.D. = S.D.
Ans: B
84. The following data gives the number of years of service of 15 employees in a manufacturing company :
5, 9, 7, 6, 24, 11, 4, 13, 10, 9, 20, 8, 19, 17, 25
The range of above data is
(a) 18 (b) 19 (c) 20 (d) 21
Ans: D
85. The diagonal elements of a skew-symmetric matrix are
(a) 1
(b) – 1
(c) 0
(d) i
Ans: C
86. If the line ax + by + c = 0 is normal to the curve xy = 1, then
(a) a > 0, b > 0
(b) a > 0, b < 0
(c) a < 0, b < 0
(d) None of these
Ans: B
87. Which of the following conditions is/are used in simplex method ?
(A) Optimality
(B) Feasibility
(a) Only (A)
(b) Only (B)
(c) Both (A) and (B)
(d) Either (A) or (B)
Ans: C
88. Which of the following interpolation formulae can be used for equal and unequal intervals ?
(a) Newton-Gregory formula (b) Bessel’s formula (c) Stirling’s formula (d) Lagrange’s formula
Ans: D
89. The equation of the plane that bisects the line segment joining the points (1, 2, 3) and (3, 4, 5) at right angle, is :
1) 2x + y + 4z = 9
2) 2x y 4z = 9
3) 2x y + 4z = 9
4) 2x + y 4z = 9
Ans: 4
90. The equation of plane passing through the intersection of the planes 4x y + z = 10 and x + y z = 4 and parallel to the line with direction ratios 2, 1, 1, is :
1) 5y 5z +6 = 0
2) 5x 5z +6 = 0
3) 5y 5z 6 = 0
4) 5x 5z 6 = 0
Ans: 3
91. Centre of Earth attracts a body with force (a) Out side surface varying as the square of distance from centre (b) Inside surface varying inversely as the distance from centre.
1) Only (a) is true
2) Only (b) is true
3) both (a) and (b) are true
4) Neither (a) nor (b) is true
Ans: 4
92. The image of x=constant under the transformation w = sinz is :
1) an ellipse 2) a hyperbola
3) a parabola 4) a circle
Ans: 2
93. The number of generators of a cyclic group of order eight is :
1) 1 2) 2
3) 3 4) 4
Ans: 4
94. If f(z) is analytic constant function in the domain D, then:
1) Only R[f(z)] is constant
2) Only I[f(z)] is constant
3) both R[f(z)]and I[f(z)] are constant
4) None of these
Ans: 3
95. Three statements are:
(a) Every field is an integral domain
(b) A field has no zero divisor,
(c) A skew field has no zero divisor. Then :
1) (a) and (b) are true but (c) is not
2) (a) and (c) are true but (b) is not
3) (b) and (c) are true but (a) is not
4) (a) (b) and (c) are true
96. The polynomial ring Z[x] over the ring of integers is a:
1) Prime ideal but not a maximum ideal
2) Maximal ideal but not a prime ideal
3) Both prime and maximal ideal
4) Commutative ring with unity only
Ans: 1
97. Let R be an integral domain and f(x) is any polymomial in R[x] with degree f(x) =r, Then f(x) is inreducible, when r is equal to :
1) 1
2) 2
3) 3
4) 4
Ans: 1
98. Let G be a group of permutations defined on a set S = {1, 2, 3, 4, 5}. Then order of proper normal subgroup of G is :
1) 30
2) 60
3) 120
4) 240
Ans: 2
99. Let AX = B is the matrix form of a system of linear equations. If rank of A=m, rank of augmented matrix [A : B] = p and number of unknowns =n, then the system has unique solution when :
1) m = p only
2) m = n = p
3) n > m or n > p
4) n > m and n > p
Ans: 2
A:-`x` + `x^(2)/2` + `x^(3)/6` B:-`x^(2)` + `x^(4)/2` + `x^(6)/6` C:-`x` + `x^(2)/2 + x^(8)/6` D:-None of these
Ans: B
52:-If F (a, b, c, x) is the hyper geometric series, then`lim_(b->oo)` F(a, b, a, `x/b` ) is
A:-`(1+x)^(a)` B:-log(1 + x) C:-`e^(x)` D:-sin x
Ans: C
53:-If `P_(n)(x)`is the Legendre Polynomial, then `P_(n)(-1)` is
A:-1 B:-− 1 C:-`(-1)^(n)` D:-0
Ans: C
54:-The equation `U_(xxxx_)+x^(2)U_(yy)` = 0 is
A:-elliptic B:-parabolic C:-hyperbolic D:-none of these
Ans: A
55:-If a, b, c are three constants such that `U_(x)` = a, `U_(y)` = b and `U_(z)` = c with u(x, y, z) = 0 at (0, 0, 0); then U(1, 0, 0) is
A:-a B:-b C:-c D:-none of these
Ans: A 8
56-If 0 < s < 1, then `int_0^oo(x^(s-1))/(1+x)dx` is
A:-`beta` (1, s) B:-`beta` (1− s, 1) C:-`beta` (1− s, s) D:-divergent
Ans: C
57:-The Laplace transform L {`(sin omegat)/(t)`} is
A:-`cot^(-1)((s)/(w))` B:-`(omega)/(s^(2)+omega^(2))` C:-`(s)/(s^(2)+omega^(2))` D:-none of these
Ans: A
58:-Which of the following is a periodic function on `RR` ?
A:-x −[x] B:-x C:-[x] D:-none of these
Ans: A
59:-Consider the fuzzy sets A and B on X = [0,`(pi)/(2)` ] where A(x) = sin x and B(x) = cos x. Then 0.8 − cut of A`nn`B is
A:-`Phi` B:-X
C:-An uncountable subset of X D:-A denumerable subset of X
Ans: A 9
60:-Which is false about the Cantor set ?
A:-Cantor set is perfect B:-Cantor set is compact
C:-Cantor set is a fractal D:-Cantor set is an uncountable set with positive Lebesgue measure
Ans: D
61:-The Voltra integral equation corresponding to the differential equation y” + xy = 1, y(0) = y’(0) = 0 is
A:-`y(x)` =`(x^(2))/(2)` −`int_0^xty(t)dt`
B:-`y(x)=x^(2)/2` −`int_0^x(x-t) y(t) dt`
C:-`y(x)=(x^(2))/(2)` −`int_0^(x)(x-t) ty (t) dt`
D:-`y(x)=(x)/(2)` −`int_0^x(x-t)ty(t) dt`
Ans: C
62:-For the Euler's gamma function Γ(z)Γ(1− z) is
A:-`(pi)/(cospiz)` B:-`(pi)/(sinpiz)` C:-`(pi)/(cotpiz)` D:-none of these
Ans: B
63:-Which of the following is not a maximal geodesic for the cylinder `x_(1)^(2)` + `x_(2)^(2)` = 1 in `RR^(3)` ?
A:-Vertical line B:-Horizontal circle C:-Helix D:-None of these
Ans: D
64:-Which of the following is reducible over `RR` ?
A:-`x^(2)` + 1 B:-`x^(2)` + 2 C:-`x^(5)` +`x^(3)+``x^(2)` + x + 1 D:-none of these
Ans: C
65. The average of 14, 11, 8, x, 19 is 13. The value of x is ______
(A) 52 (B) 12 (C) 16 (D) 13
Ans: D
66. If a : b = 2 : 3, b : c = 4 : 5, then a : b : c = ______
(A) 4 : 3 : 5 (B) 2 : 4 : 5 (C) 8 : 12 : 15 (D) 8 : 10 : 20
Ans: C
67. If 20% of a number is 140, then 16% of that number is _____
(A) 120 (B) 114 (C) 110 (D) 112
Ans: D
68. If by selling an article for Rs. 360, Ramu gains 20%. Find his cost price.
(A) 320 (B) 300 (C) 310 (D) 380
Ans: B
69:-16×8(6 − 4) ÷2+9= ___________
A:-219 B:-160 C:-137 D:-90
Ans: C
70:-20.4 × 0.44 ÷ 3.52 = ___________
A:-2.55 B:-3.55 C:-2.58 D:-5.01
Ans: A
71:-The average of 10 numbers is 30. If the average of 6 of these numbers is 29, what is the average of the remaining 4 numbers ? A:-31 B:-28.5 C:-30 D:-31.5
Ans: D
72:-The ratio of the length of a ground to its width is 5 : 2. Find its length if the width is 50 meters
A:-100 m B:-125 m C:-160 m D:-120 m
Ans: B
73:-A sum of money deposited at 2% per annum compounded annually becomes Rs. 10,404 at the end of 2 years. Find the sum deposited A:-Rs. 10,000 B:-Rs. 10,400 C:-Rs. 8,000 D:-Rs. 9,500
Ans: A
74:-40 men can do a piece of work in 20 days. If there are 50 men to do the same work how long will they take to finish it ?
A:-15 days B:-20 days C:-16 days D:-10 days
Ans: C
75:-A train running at 54 Km/hr crosses a telegraph post in 10 seconds. What is the length of the train ?
A:-540 m B:-150 m C:-100 m D:-75 m
Ans: B
76. A relation R = {(1, 1), (1, 2), (2, 1)} is defined on the set A = {1, 2, 3}, then relation is
(a) Reflexive (b) Reflexive and transitive (c) Equivalence relation (d) Symmetric
Ans: D
77. The number of generators of a cyclic group of order 12 is
(a) 4 (b) 3 (c) 2 (d) 1
Ans: A
78. 20 persons were invited for a party. What is the number of ways in which they and the host can be seated at a circular table such that two particular persons be seated on either side of the host ?
(a) 20 ! (b) 19 ! (c) 2(18 !) (d) (18 !)
Ans: C
79. The two positive numbers, whose difference is 12 and whose A.M. exceeds their G.M. by 2, are
(a) 32, 20 (b) 25, 13 (c) 20, 8 (d) 16, 4
Ans: D
80. How many unrelated conditions are required to determine a plane ?
(a) 4 (b) 3 (c) 2 (d) 1
Ans: B
81. Which of the following is the best measure of dispersion ?
(a) Range (b) Mean deviation (c) Standard deviation (d) Co-efficient of variation
Ans: C
82. Probability that a leap year contains 53 Sundays is
(a) 1/ 7
(b) 2 /7
(c) 3/ 7
(d) 4/ 7
Ans: B
83. The relationship between mean deviation (M.D.) and the standard deviation (S.D.) in the normal distribution, is approximately
(a) 3 M.D. = 2 S.D. (b) 5 M.D. = 4 S.D. (c) 6 M.D. = 5 S.D. (d) M.D. = S.D.
Ans: B
84. The following data gives the number of years of service of 15 employees in a manufacturing company :
5, 9, 7, 6, 24, 11, 4, 13, 10, 9, 20, 8, 19, 17, 25
The range of above data is
(a) 18 (b) 19 (c) 20 (d) 21
Ans: D
85. The diagonal elements of a skew-symmetric matrix are
(a) 1
(b) – 1
(c) 0
(d) i
Ans: C
86. If the line ax + by + c = 0 is normal to the curve xy = 1, then
(a) a > 0, b > 0
(b) a > 0, b < 0
(c) a < 0, b < 0
(d) None of these
Ans: B
87. Which of the following conditions is/are used in simplex method ?
(A) Optimality
(B) Feasibility
(a) Only (A)
(b) Only (B)
(c) Both (A) and (B)
(d) Either (A) or (B)
Ans: C
88. Which of the following interpolation formulae can be used for equal and unequal intervals ?
(a) Newton-Gregory formula (b) Bessel’s formula (c) Stirling’s formula (d) Lagrange’s formula
Ans: D
89. The equation of the plane that bisects the line segment joining the points (1, 2, 3) and (3, 4, 5) at right angle, is :
1) 2x + y + 4z = 9
2) 2x y 4z = 9
3) 2x y + 4z = 9
4) 2x + y 4z = 9
Ans: 4
90. The equation of plane passing through the intersection of the planes 4x y + z = 10 and x + y z = 4 and parallel to the line with direction ratios 2, 1, 1, is :
1) 5y 5z +6 = 0
2) 5x 5z +6 = 0
3) 5y 5z 6 = 0
4) 5x 5z 6 = 0
Ans: 3
91. Centre of Earth attracts a body with force (a) Out side surface varying as the square of distance from centre (b) Inside surface varying inversely as the distance from centre.
1) Only (a) is true
2) Only (b) is true
3) both (a) and (b) are true
4) Neither (a) nor (b) is true
Ans: 4
92. The image of x=constant under the transformation w = sinz is :
1) an ellipse 2) a hyperbola
3) a parabola 4) a circle
Ans: 2
93. The number of generators of a cyclic group of order eight is :
1) 1 2) 2
3) 3 4) 4
Ans: 4
94. If f(z) is analytic constant function in the domain D, then:
1) Only R[f(z)] is constant
2) Only I[f(z)] is constant
3) both R[f(z)]and I[f(z)] are constant
4) None of these
Ans: 3
95. Three statements are:
(a) Every field is an integral domain
(b) A field has no zero divisor,
(c) A skew field has no zero divisor. Then :
1) (a) and (b) are true but (c) is not
2) (a) and (c) are true but (b) is not
3) (b) and (c) are true but (a) is not
4) (a) (b) and (c) are true
96. The polynomial ring Z[x] over the ring of integers is a:
1) Prime ideal but not a maximum ideal
2) Maximal ideal but not a prime ideal
3) Both prime and maximal ideal
4) Commutative ring with unity only
Ans: 1
97. Let R be an integral domain and f(x) is any polymomial in R[x] with degree f(x) =r, Then f(x) is inreducible, when r is equal to :
1) 1
2) 2
3) 3
4) 4
Ans: 1
98. Let G be a group of permutations defined on a set S = {1, 2, 3, 4, 5}. Then order of proper normal subgroup of G is :
1) 30
2) 60
3) 120
4) 240
Ans: 2
99. Let AX = B is the matrix form of a system of linear equations. If rank of A=m, rank of augmented matrix [A : B] = p and number of unknowns =n, then the system has unique solution when :
1) m = p only
2) m = n = p
3) n > m or n > p
4) n > m and n > p
Ans: 2